An Overview of Gradient-Enhanced Metamodels with Applications

Metamodeling, the science of modeling functions observed at a finite number of points, benefits from all auxiliary information it can account for. Function gradients are a common auxiliary information and are useful for predicting functions with locally changing behaviors. This article is a review of the main metamodels that use function gradients in addition to function values. The goal of the article is to give the reader both an overview of the principles involved in gradient-enhanced metamodels while also providing insightful formulations. The following metamodels have gradient-enhanced versions in the literature and are reviewed here: classical, weighted and moving least squares, Shepard weighting functions, and the kernel-based methods that are radial basis functions, kriging and support vector machines. The methods are set in a common framework of linear combinations between a priori chosen functions and coefficients that depend on the observations. The characteristics common to all kernel-based approaches are underlined. A new -GSVR metamodel which uses gradients is given. Numerical comparisons of the metamodels are carried out for approximating analytical test functions. The experiments are replicable, as they are performed with an opensource available toolbox. The results indicate that there is a trade-off between the better computing time of least squares methods and the larger versatility of kernel-based approaches.